Such a method is known from ZHANG, H.: Triple frequency Cascading Ambiguity Resolution for Modernized GPS and Galileo. In: UCGE Reports, No. 20228, 2005. According to the known method three frequency linear combinations for Galileo are proposed. A stringent bound on the weighting coefficients, however, prevented the generation of new widelane combinations above 0.90 m.
Currently three global navigation satellite systems are used or proposed: The global positioning system (GPS), Glonass, and the proposed Galileo system. The global navigation satellite systems are based on navigation satellites which emit carrier signals in the higher frequency range. A periodical code sequence and a navigation message are modulated on the carrier signals. Based on the code sequence and the navigation message a positioning process of a navigation device can be performed. The accuracy of the positioning can considerably be enhanced if the high frequency carrier signal is used for the positioning process besides the code signal. For instance, within GPS, the wavelength of the carrier signal L1 amounts to 19.0 cm compared to 300 m of a code chip. In consequence an accuracy improvement by a factor of 1500 is to be expected.
One disadvantage of a positioning process using the carrier signals is the ambiguity of the phase since the carrier signal contains no information on the integer number of wavelengths between the navigation device and the satellite. Therefore, the phasing is unknown.
Furthermore, it has to be taken into account, that the carrier signal pass through the earth's atmosphere particular through the ionosphere. In consequence the carrier signals show a so called ionospheric error. Further error sources are the phase noise of the carrier signal, the orbital error of the satellite as well as the clock error of the navigation device and the satellite.
The clock errors can be eliminated by the double difference method in which the position of the navigation device with respect to a reference station is determined by considering the differences between current difference signals from different satellites, wherein the difference signals are formed by the differences between the signals emitted by a specific satellite but received from the navigation device and the reference station. In addition ionospheric and tropospheric errors are significantly attenuated for short baselines if the double difference method is used.
In general, the phase ambiguity is resolved by estimating the phasing. The reliability of the estimate depends among other things on the relation between the wavelength of the carrier signal and the deviations resulting from the other error sources. Typically, the ionospheric error results in a spatial error of a few meters, whereas the phase noise affects the results of the phase estimation by a few millimeters.
The carrier phase measurements are highly accurate but ambiguous measurements. Numerous approaches have been suggested for integer ambiguity resolution.
TEUNISSEN, P.: Least-squares estimation of the integer ambiguities, Invited lecture, Section Ito IV, Theory and Methodology, IAG General Meeting, Beijing, China, 1993 discloses a least square estimation method for resolving phase ambiguities. This approach is also called the LAMBDA(=Leastsquares Anbiguity Decorrelation Adjustment)-method.
DE JONGE, D. and TIBERIUS, C.: The LAMBDA method for integer ambiguity estimation: implementation aspects In: LGR series, Delft University of Technology, pp. 1-49, 1996 discloses further details on the implementation of a least square estimation method for resolving phase ambiguities.
Details on methods for resolving the phase ambiguity can also be found in HENKEL, P. and GÜNTHER, C.: Integrity Analysis of Cascade Integer Resolution with Decorrelation Transformations. In: Proceedings of the Institute of Navigation, National Technical Meeting, San Diego, 2007 and in US 2005/101248 A1.
The reliability of integer estimation is validated by the success rate disclosed in TEUNISSEN, P.: Success probability of integer GPS ambiguity rounding and bootstrapping. In: Journal of Geodesy, Vol. 72, pp. 606-612, 1998. or by a comparison between the error norms of the best and second-best integer candidates as disclosed in VERHAGEN, S.: On the Reliability of Integer Ambiguity Resolution. In: Journal of the Institute of Navigation, Vol. 52, No. 2, pp. 99-110, 2005.
By combining carrier signals a combined signal can be generated which comprises a significantly greater wavelength than the single carrier signals. For instance, the difference between the carrier signals L1 and L2 results in a combined signal with a wavelength of 86.2 cm, which is therefore referred to as widelane (WL). The sum of the carrier signals L1 and L2 results in a combined signal with a wavelength of 10.7 cm, which is referred to as narrowlane (NL).
A systematic search of all possible widelane combinations of L1 and L2 is disclosed in COCARD, M. and GEIGER, A.: Systematic search for all possible widelanes. In: Proc. of 6th Intern. Geodetic Symposium on Satellite Positioning, 1992. The widelane combination and the narrowlane combinations are characterized by the noise amplification, the amplification of the ionospheric error and by the amplification of the multipath error.
In COLLINS, P.: An overview of GPS inter-frequency carrier phase combinations. In: UNB/GSD, 1999 a method for a systematic search for GPS inter-frequency carrier phase combinations is disclosed. Ionospheric, noise and multipath characteristics are computed for both widelane and narrowlane L1-L2 combinations.
VOLLATH, U. et. Al: Three or Four Frequencies-How many are enough? In: Proc. of the Institute of Navigation (ION), Portland, USA, 2000 describes the reliability of ambiguity resolution for navigation systems with four carriers.
For resolving the phase ambiguity an iterative approach is used. In the beginning, the phase is estimated using a combined signal with the greatest wavelength. In subsequent iteration steps, further combined signals with stepwise decreasing wavelength are considered. In each iteration step the phasing is estimated wherein the information on the phasing gained in previous iteration steps can be used in subsequent iteration steps. This approach is often referred to as CIR(=Cascade Integer ambiguity Resolution). The cascade integer ambiguity resolution is based on different widelane combinations. The integer estimation simplifies with increased wavelength which is of special interest in Wide Area Real-Time Kinematics (WARTK) due to different ionospheric conditions at the user and reference location.
In JUNG, J.: High Integrity Carrier Phase Navigation Using Multiple Civil GPS Signals. In: Ph.D. Thesis, University of Stanford, 2000 three frequency linear combinations for GPS and extended CIR are analyzed by ionospheric gradient estimation.
The disadvantage of the prior art cascade integer ambiguity resolution is the repetitive estimation of the baseline which is a nuisance parameter in all steps except the last one. This motivates the use of geometry-free widelane combinations.
The most simple geometry-free combinations are obtained by subtracting phase measurements of two different frequencies. The E5a-E5b combination benefits from a significant reduction of the ionospheric error but shows a severe drawback: The superposition of ambiguities can no longer be expressed as an integer multiple of a single wavelength.
SIMSKY, A.; SLEEWAEGEN, J.-M. and NEMREY, P.: Early performance results for new Galileo and GPS signals-in-space, European Navigation Conference, describe a so called multipath-combination which removes geometry portion, ionospheric and tropospheric errors but suffers again from the loss of integer nature.
KAPLAN, E.; HEGARTY, C.: Understanding GPS—Principles and Applications, Artech House, 2nd edition, Norwood (Mass.), 2006 and European Space Agency (ESA) and Galileo Joint Undertaking (GJU), GAL OS SIS ICD, May 2006 contains considerations on possible error sources.